In calculus, **differentiation** is one of the two important concept apart from integration. It is a method of finding the derivative of a function or instantaneous rate of change in function based on one of its variables. If x is a variable and y is another variable, then the rate of change of x with respect to y is given by dy/dx.

Functions are generally classified in two categories under Calculus, namely:

**(i) Linear functions**

** (ii) Non-linear functions**

A linear function varies with a constant rate through its domain. Therefore, the overall rate of change of the function is the same as the rate of change of a function at any point.

However, the rate of change of function varies from point to point in case of non-linear functions. The nature of variation is based on the nature of the function.

The rate of change of a function at a particular point is defined as a **derivative** of that particular function.

## Differentiation in Calculus

Differentiation, in terms of calculus, can be defined as a derivative of a function regarding the independent variable and can be applied to measure the function per unit change in the independent variable.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:

**dy / dx**

If the function f(x) undergoes an infinitesimal change of h near to any point x, then the derivative of the function is defined as

\(\lim\limits_{h \to 0} \frac{f(x+h) – f(x)}{h}\)

When a function is denoted as y=f(x), the derivative is indicated by the following notations.

**D(y) or D[f(x)]**is called Eulerâ€™s notation.**dy/dx**is called Leibnizâ€™s notation.**Fâ€™(x)**is called Lagrangeâ€™s notation.

Differentiation is the process of determining the derivative of a function at any point.

## Differentiation Formulas

Some of the important Differentiation formulas in differentiation are as follows.

- If f(x) = tan (x), then f'(x) = sec
^{2}x - If f(x) = cos (x), then f'(x) = -sin x
- If f(x) = sin (x), then f'(x) = cos x
- If f(x) = ln(x), then f'(x) = 1/x
- If f(x) = \(e^{x}\), then f'(x) = \(e^{x}\)
- If f(x) = \(x^{n}\), where n is any fraction or integer, then f'(x) = \(nx^{n-1}\)
- If f(x) = k, where k is a constant, then f'(x) = 0

**Also, see:**

### Differentiation Rules

Some of the basic differentiation rules that need to be followed are as follows.

#### (i) Sum or Difference Rule

If the function is sum or difference of two functions, the derivative of the functions is the sum or difference of the individual functions, i.e.,

**If f(x)=u(x)Â±v(x)**

**then, f'(x)=u'(x)Â±v'(x)**

#### (ii) Product Rule

As per the product rule, if the function f(x) is product of two functions u(x) and v(x), the derivative of the function is,

**If \(f(x) = u(x) \times v(x)\)**

**then, \(\mathbf { f'(x) = u'(x) \times v(x) + u(x) \times v'(x)}\)**

#### (iii) Quotient rule

If the function f(x) is in the form of two functions [u(x)]/[v(x)], the derivative of the function is

**If, \(f(x) = \frac{u(x)}{v(x)}\)**

**then, \(\large \mathbf { f'(x) = \frac{u'(x) \times v(x) – u(x) \times v'(x)}{(v(x))^{2}}}\)**

#### (iv) Chain Rule

If a function y = f(x) = g(u) and if u = h(x), then the chain rule for differentiation is defined as,

\(\large \mathbf{\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\mathrm{d} y}{\mathrm{d} u} \times \frac{\mathrm{d}u }{\mathrm{d} x}}\)

This plays a major role in the method of substitution that helps to perform differentiation of composite functions.

### Differentiation Examples

With the help of differentiation, we are able to find the rate of change of one quantity with respect to another. Some of the examples are:

- Acceleration: Rate of change of velocity with respect to time
- To calculate the highest and lowest point of the curve in a graph or to know its turning point, the derivative function is used
- To find tangent and normal to a curve

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